1. Antiderivatives as Areas
OBJECTIVE
Find the area under a graph to solve real-world problems
Use rectangles to approximate the area under a graph.
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2. 4.2 Antiderivatives as Areas
Example 1: A vehicle travels at 50 mi/hr for 2 hr. How far has the vehicle traveled?
The answer is 100 mi. We treat the vehicle’s velocity as a function, We graph this function, sketch a vertical line at and obtain a rectangle. This rectangle measures 2 units horizontally and 50 units vertically. Its area is the distance the vehicle has traveled:
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3. 4.2 Antiderivatives as Areas
Example 2: The velocity of a moving object is given by the function where x is in hours and v is in miles per hour. Use geometry to find the area under the graph, which is the distance the object has traveled:
a.) during the first 3 hr
b.) between the third hour and the fifth hour
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4. 4.2 Antiderivatives as Areas
Example 2 (continued):
a.) The graph of the velocity function is shown at the right. We see the region corresponding to the time interval is a triangle with base 3 and height 9 (since ). Therefore, the area of this region is
The object traveled 13.5 mi during the first 3 hr.
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5. 4.2 Antiderivatives as Areas
Example 2 (Continued):
b.) The region corresponding to the time interval is a trapezoid. It can be decomposed into a rectangle and a triangle as indicated in the figure to the right. The rectangle has a base 2 and height 9, and thus an area
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6. 4.2 Antiderivatives as Areas
Example 2 (Concluded):
b.) The triangle has base 2 and height 6, for an area
Summing the two areas, we get 24. Therefore, the object traveled 24 mi between the third hour and the fifth hour.
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7. 4.2 Antiderivatives as Areas
Quick Check 1
An object moves with a velocity of where t is in minutes and v is in feet per minute.
a.) How far does the object travel during the first 30 min?
b.) How far does the object travel between the first hour and the second hour?
a.) We know that this is a linear function, so the region corresponding to the time interval is a triangle with base 30 and height 15 (since ). Therefore the area of this region is
The object traveled 225 feet in the first 30 minutes.
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8. 4.2 Antiderivatives as Areas
Quick Check 1 Concluded
b.) How far does the object travel between the first hour and the second hour?
The region corresponding to the time interval is a trapezoid. It can be decomposed into a rectangle and a triangle. The rectangle has base 60 and height 30 (since ). The triangle is has base 60 and height 30 (since ). Therefore the area of this region is Thus the object traveled 2,700 feet between the first hour and second hour.
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9. 4.2 Antiderivatives as Areas
Riemann Sums:
The last two examples, the area function is an antiderivative of the function that generated the graph. Is this always true? Is the formula for the area under the graph of any function that function’s antiderivative? How do we handled curved graphs for which area formulas may not be known? We investigate the questions using geometry, in a procedure called Riemann summation.
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10. 4.2 Antiderivatives as Areas
Riemann Sums (continued):
In the following figure, [a, b] is divided into four subintervals, each having width x = (b – a)/4.
The heights of the rectangles are f (x1), f (x2), f (x3) and f (x4).
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11. 4.2 Antiderivatives as Areas
Riemann Sums (concluded):
The area of the region under the curve is approximately the sum of the areas of the four rectangles:
We can denote this sum with summation, or sigma, notation, which uses the Greek capital letter sigma, or 
This is read “the sum of the product from
” To recover the original expression, we substitute the numbers 1 through 4 successively for i in and write plus signs between the results.
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12. 4.2 Antiderivatives as Areas
Example 3: Write summation notation for
Note that we are adding consecutive values of 2.
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13. 4.2 Antiderivatives as Areas
Quick Check 2
Write the summation notation for each expression
a.)
b.)
Note that we are adding consecutive multiples of 5. Thus,
Note that we are adding consecutive multiples of 11. Thus,
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14. 4.2 Antiderivatives as Areas
Example 4: Write summation notation for:
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15. 4.2 Antiderivatives as Areas
Example 5: Express without using summation
notation.
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16. 4.2 Antiderivatives as Areas
Quick Check 3
Express without using summation notation.
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17. 4.2 Antiderivatives as Areas
Example 6: Express without using
summation notation.
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18. 4.2 Antiderivatives as Areas
Example 7: Consider the graph of:
over the interval [0, 600].
a) Approximate the area by dividing the interval into 6 subintervals.
b) Approximate the area by dividing the interval into 12 subintervals.
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19. 4.2 Antiderivatives as Areas
Example 7 (continued):
a) We divide [0, 600] into
6 intervals of size
with xi ranging from x1 = 0
to x6 = 500.
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20. 4.2 Antiderivatives as Areas
Example 7 (continued):
Thus, the area under the curve is approximately
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21. 4.2 Antiderivatives as Areas
Example 7 (continued):
b) We divide [0, 600] into
12 intervals of size
with xi ranging from x1 = 0
to x12 = 550.
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22. 4.2 Antiderivatives as Areas
Example 7 (concluded):
Thus, the area under the curve is approximately
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23. 4.2 Antiderivatives as Areas
Section Summary
The area under a curve can often be interpreted in a meaningful way.
The units of the area are found by multiplying the units of the input variable by the units of the output variable. It is crucial that the units are consistent.
Geometry can be used to find areas of regions formed by graphs of linear functions.
A Riemann sum uses rectangles to approximate the area under a curve. The more rectangles, the better approximation.
The definite integral is a representation of the exact area under the graph of a continuous function where over an interval
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